### Ultracategories and 2-Monads

#### Posted by David Corfield

We began a discussion of Jacob Lurie’s Ultracategories over here, in particular whether they may be construed as algebras for some 2-monad. Perhaps this topic deserves a post to itself, rather than appearing tucked at the end of a long and fascinating discussion about condensed/pyknotic mathematics.

I have just discovered a 1995 PhD thesis by Francisco Marmolejo, advised by Robert Paré, that’s very relevant. Unfortunately the only online access is to a very poor photocopy, here. Anyway, Marmolejo characterises Makkai’s ultracategories there in a 2-monadic way. This would still leave Lurie’s somewhat differently defined ultracategories on the to-do list.

There’s then a follow-up question of characterising any such 2-monads using the codensity monad construction if possible. There’s some ongoing codensity conversation over here.

This area isn’t just for categorical logicians – some philosophers are paying attention, see

- Halvorson, Hans and Tsementzis, Dimitris (2015),
*Categories of scientific theories*, (preprint)

especially pp. 20-21.

…there seems to be nothing inherently “category-theoretic” about ultraproducts. As Makkai’s work proves and Los’ theorem has long made obvious, taking ultraproducts is a fundamental operation when it comes to elementary classes: elementary classes are exactly those classes closed under elementary equivalence and the taking of ultraproducts. Since every pretopos corresponds to an elementary class (more precisely: to the category of models of a coherent theory) one would imagine that any such characterization of

Pretopwould amount to a characterization of “closure under ultraproducts”. Absent any useful purely categorical description of ultraproducts (or even ultrafilters) this seems like a significant obstruction. Nevertheless the work of Leinster (2013) on ultrafilter monads as codensity monads might provide a way out, though this is still very far from being made precise.

Finally, I still have an open question about how to characterise the $\infty$-version developed by Lurie as a logical result about syntax-semantic duality. What kind of ‘theory’ is involved?

## Re: Ultracategories and 2-Monads

Section 4.5.3 of Marmolejo’s thesis tells us of a 2-adjunction between $PRETOP^{op}$ and $CAT$, with 2-functors $Mod(-)$ and $Set^{(-)}$. Given an algebra for the resulting 2-monad, $T$, on $CAT$, that is, a category, $A$, and functor $\Phi: Mod(Set^A) \to A$, then given an ultrafilter $(I, U)$, we can form an ultraproduct in $A$ via the ultraproduct map, $Mod(Set^A)^I \to Mod(Set^A)$.

This gives us a pre-ultracategory. Then a further 2-adjunction between $PRETOP^{op}$ and $T-ALG$ is used to generate a 2-monad whose algebras give Makkai ultracategories.